Eur. Phys. J. B 13, 745–754 (2000) THE EUROPEANPHYSICAL JOURNAL Bc©

EDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2000

Thermoelastic behaviour of polyvinylacetate monolayers atthe air-water interface: Evidences for liquid-solid phase transition

F. Monroy1,a, F. Ortega2, and R.G. Rubio2

1 Laboratoire de Physique des Solides, Batiment 510, Universite Paris-Sud, 91405 Orsay Cedex, France2 Departamento de Quımica Fısica I, Facultad de Ciencias Quımicas, Universidad Complutense, 28040 Madrid, Spain

Received 12 January 1999 and Received in final form 11 June 1999

Abstract. The thermoelastic behaviour of polyvinylacetate monolayers spread on an aqueous subphase hasbeen studied using rheological data previously published (Monroy et al., Phys. Rev. E 58, 7629 (1998)).The results show fluid-like viscoelastic behaviour well above a transition temperature Tm, while at lowertemperatures a soft solid-like behaviour emerges. The correlation between thermodynamic and elasticproperties below Tm can be described in terms of scaling laws.

PACS. 68.10.-m Fluid surfaces and fluid-fluid interfaces – 68.10.Et Interface elasticity, viscosity, andviscoelasticity – 68.60.-p Physical properties of thin films, nonelectronic – 82.65.Dp. Thermodynamics ofsurfaces and interfaces

1 Introduction

Monomolecular films of small surfactants located at theair-water interface are known to form various solid phases,which have been extensively characterised by differentdiffraction techniques, particularly grazing incident X-rayfrom synchrotron sources [1] or neutron diffraction [2]. Onthe other hand, these quasi-bidimensional systems melt atlow surface density and high enough temperature, appear-ing different fluid phases. In contrast, the low-temperaturephases found for monolayers of insoluble polymers do notshow, in most cases, crystalline order. In fact, in 3D it iswell-known that polymeric systems only form well-definedsolid states for some specific tacticities [3]. The presence ofchain defects in atactic polymers or polymers with lateralbranches, makes it impossible for the polymers to crys-tallize on cooling, forming amorphous glasses or at mostsemicrystalline solids. It is commonly accepted that thetwo generic types of polymeric solids, namely crystallineand glassy polymers, give rise to different elastic states,energy-driven and entropy-driven (rubber-like elasticity),respectively. In the first case, at small strains the displace-ment of the elastic elements causes a variation in internalenergy around the energetic minimum, which correspondsto the equilibrium arrangement in the crystalline lattice,while the entropy remains approximately constant. How-ever, a rubber decreases its internal entropy on elonga-tion, reason for which it recovers its equilibrium dimen-sion if the applied stress ceases. Crystalline order has been

a Permanent address: Dept. Quımica Fısica, Univ.Complutense Madrid.e-mail: monroy@eucmos.sim.ucm.es

however found by Schouten et al. [4] in monolayers ofPMMA on water if the polymer have a certain degree oftacticity, while the glassy state is the only kinetically ac-cessible phase for the atactic form, as in the 3D system.Furthermore, because of the increase in chain mobility inthe monolayer, the loss of a degree of freedom will de-crease Tg with regards to the 3D value [5,6]. Therefore,polymer concentrated solutions in 2D are excellent candi-dates to check for fluid-solid transitions in 2D, because,though being basically liquids, normally they behave likehighly non-Newtonian viscoelastic fluids, and also undercertain circumstances, they seem to behave like a purelyelastic solid.

The rheological criterion for distinguishing between asolid and a fluid is that the first exhibits resistance to sheardeformation because of the bonding between defects in thesolid lattice. In the liquid state at a high enough temper-ature, it is favourable for the bounded pairs of defects todissociate because of the resulting gain in entropy. At thispoint free dislocations appear and the system will respondto a shear stress with no resistance, the mechanism beingthe continuous creation of dislocation pairs, and pullingthem apart to infinity. In these conditions shearing doesnot store energy in elastic form, only dissipation is present,and we have viscous shear flow [7].

On the other hand, it is well-understood that thecharacter of ordering present in a system depends onits spatial dimensionality [8]. Indeed, it is also generallyaccepted that because of fluctuations, solids in one ortwo dimensions lack true long-range order, therefore themelting phenomena is not an expected feature in thesesystems [8,9]. However, and based in a new definition of

746 The European Physical Journal B

order, so-called “topological order”, a theoretical frame for2D-freezing has been developed by Kosterlitz–Thouless–Halperin–Nelson–Young [10,11]. In the KTHNY theory,the loss of the long-range topological order, renders amelting behaviour essentially different to the tridimen-sional case. Two-dimensional solids melt via sequentialcontinuous phase transition through a hexatic interme-diate phase with short-range positional order and quasi-long range bond orientational order. It is also commonlyaccepted that KTHNY theory of 2D-melting is not basedon any particular choice of intermolecular interactions; itremains valid for any system, which can be characterisedas a deformable elastic medium [12].

Fluid polymer monolayers [13] can be consideredas complex quasi-bidimensional systems, with negligiblethickness h, compared with planar dimensions. Hence theybehave as very stiff membranes characterised by a set of2D viscoelastic parameters several orders of magnitudelarger than the analogue ones for the bulk system. Partic-ularly, previous studies have demonstrated that polyviny-lacetate (PVAc) spread at the air-water interface is ableto form very stable monolayers [14–16]. The PVAc at-actic chains present extended configurations because ofthe good-solvent character of the air-water interface forthis polymer, which produces a thermodynamic behaviourcharacterised by very expanded isotherms [14–17]. In aprevious paper [18], we have carried out a rheologicalstudy of these monolayers in a broad frequency range(> 10 decades) and between 0 and 25 ◦C. To thisend, quasi-elastic surface light scattering (QESLS), ex-cited electrocapillary waves (ECW) and mechanical step-compression experiments were performed.

We have pointed out the strong correlation betweenthe static thermoelastic properties and the dynamic ones,evidencing a high elastic efficiency, which is better at lowertemperatures. In addition, a singular behaviour was de-tected at a temperature around 14 ◦C, which might becompatible with a transition from a fluid to a soft-solidstate. All these issues make this system interesting forchecking fluid-solid transitional behaviour in 2D polymericmatter. The results concerning the surface rheology of thePVAc monolayers have been previously described in ref-erence [18]. However, the results were only described froma phenomenological point of view, i.e., no microscopic ap-proach was proposed to explain the observed macroscopicbehaviour, unless a chain reptation model was tentativelyproposed to explain the complex dynamics found in thelow frequency domain. From these data and at surfaceconcentration high enough, the monolayer seems to be-have as a quasi-2D polymer gel, made of a network ofpolymer chains in a near-linear conformation.

The main aim of this work is hence twofold: First, toshow clear signatures for the existence of transition froma fluid to a soft-solid state (like a 2D rubber) in the PVAcmonolayers as the temperature is decreased. Second, topoint out the existence of singular thermoelastic behaviourcompatible with bidimensional solidification processes.

The rest of this paper will be organised as follows:Section 2 briefly summarizes the essential concepts of

surface hydrodynamics related to our experimental re-sults. Section 3 contains the thermoelastic information de-rived from the experimental results, and scaling behaviourfound for some of the elastic parameters. Finally, Section 4summarizes the main conclusions.

2 Theoretical background

It is well-known that because of the coupling betweentransversal and longitudinal motions at the air-water in-terface, surface hydrodynamics is relevant to the thermoe-lastic behaviour of polymer films [19]. The transversal orcapillary motion is essentially a shear motion, governedby surface tension γ and gravity g, while the longitudinalmotion must be decomposed in both, a pure compressionmode and a shear mode, characterised by the two differ-ent elastic moduli (λ and µ, respectively). Since dissipativeeffects do exist within the film, each one of these uniax-ial moduli must be linearly developed as a viscoelasticresponse function with imaginary part proportional to aviscosity coefficient. The bulk modulus ε, defined in equa-tion (1), contains both shear µ and compression λ, and it iscommonly named dilational modulus. This total longitudi-nal modulus contains a real part or dilational elasticity ε,and a dissipative term or dilational viscosity κ(= ηµ+ηλ).

ε(T ) = µ(T ) + λ(T ) = ε+ iωκ. (1)

Although in general the shear contribution to dilationalviscosity ηµ, presents non-zero values, it is several ordersof magnitude smaller than the compression componentηλ [20,21].

Recently, Buzza et al. [22] and earlier Seppecher [23],from rigorous thermodynamic equalities for the viscoelas-tic free energy functional, have theoretically demonstratedthat the transversal dissipation coefficient must be zero inorder to obtain a quadratic form for the surface dissipa-tive function, and consequently a minimal dissipation atnull deformation [7,22]. In spite that this treatment incor-porates new elastic constants related to the spontaneouscurvature and its coupling with surface dilatation, they be-come negligible compared to λ when the interfacial thick-ness h is small enough as compared with the wavevector,q (i.e. qh � 1). Consequently, the usual hydrodynamicmodel (see Ref. [19]) with a real and a ω–independenttransversal response function equal to the equilibrium sur-face tension, γ = γ0 [22], will be valid in the linear vis-coelastic regime and for the particular system studiedhere (qh ≤ 10−4).

In order to set the meaning of some of the vari-ables used below, it is convenient to briefly review thebackground associated to the capillary wave experiments.When the hydrodynamic Navier-Stokes equation for os-cillatory motion is resolved within the appropriate inter-facial boundary conditions, one finds capillary and dila-tional modes coupled together [19,22,24]. The dispersionequation D(ω), for this coupled motion finally can bewritten as:

D(ω) = LT + [ωη(q −m)]2 = 0 (2)

F. Monroy et al.: Thermoelastic behaviour of polyvinylacetate monolayers at the air-water interface 747

where:

L = εq2 + iωη(q +m) (3)

T = γq2 + iωη(q +m)− ρ

qω2 + ρg (4)

m is known as capillary penetration depth in the subphaseand is given by:

m2 = q2 + iωρ

ηwith Re(m) > 0. (5)

Solving the dispersion equation for γ = γ0 and for thepropagation characteristics of the surface modes ω, q, onecan obtain the dilational parameters ε = ε+ iωκ [18].

3 Results and discussion

3.1 Equilibrium thermoelasticity

The thermodynamic equilibrium properties of the PVAcmonolayers spread at the air-water interface have beenextensively discussed in detail in a previous work [18]. Wewill reanalyse here some of the thermoelastic informationthat will be used later in discussing the solidification-likebehaviour.

3.1.1 Concentration regimes and lattice dimension a0

From the experimental surface pressure-surface concen-tration (Π−Γ ) isotherms we did calculate the critical ex-ponent of the radius of gyration ν for PVAc, and foundthat it remains constant within the experimental uncer-tainty at 0.78±0.03 over the whole temperature range [16].This value is in excellent agreement with the predictionof the Renormalization Group theory for polymers in abidimensional good solvent [25]. We were also able to cal-culate the chain overlap concentration Γ ∗, which marksthe crossover from dilute to semidilute regimes [25]. Atthis particular concentration the chains begin to overlapand the lateral contact between two independent chains isestablished. Therefore, one can consider this arrangementas the more compact 2D distribution compatible with themonomer size. Consequently, the area per monomer atΓ ∗(A∗ = 1/Γ ∗) can be taken as an estimation of the di-mensions of a square lattice a2

0 ≈ A∗. Figure 1 shows thatA∗ increases strongly with T , reflecting the tendency ofthe monolayers to be more expanded at high tempera-ture. Another possible estimation of a0 can be obtainedfrom the so-called limit area of the isotherm Alim, or sur-face area of the semidilute regime extrapolated at Π = 0.In general, these values are smaller than A∗, but as it canbe seen in Figure 1, both exhibit the same tendency on in-creasing T . In these data one observes a clear kink around14−15 ◦C which suggests a change in the specific area oncooling the monolayers. This feature will be discussed laterin detail.

In order to obtain a good estimation of a0, continu-ous compression-expansion cycles in the dilute-semidilute

0 5 10 15 20 2522

24

26

40

60

80

T (°C)

A (Å

2/m

onom

er)

Fig. 1. Temperature dependence of the area per monomer atΓ ∗(= 1/A∗) (�). For comparison, the cell dimension A0 ob-tained from the Singer’s equation of state (◦) is also repre-sented. (�) Corresponds to the limit area Alim of the isothermsof the PVAc monolayers.

regime (Π < 2 mN/m) were performed in a Lang-muir trough at low enough velocity of the barriers(< 1 mm/min) [14]. No hysteresis processes were de-tected in these closed cycles. These low Π portions of theisotherms were fitted to the Singer’s equation of state (2Dversion of the configurational lattice model of Flory) cor-rected with a Van der Waals energetic contribution [26].This equation is able to describe successfully the dilutedportion of the experimental isotherms and from the fitsone can obtain a series of microscopic parameters. Partic-ularly the cell dimensions a0, and the coordination indexz of a polymer site in the lattice has been calculated as afunction of temperature. The data in Figure 1 show theexcellent agreement of the cell parameter a0 =

√A0 calcu-

lated from these fits to the equation of state and the areaper monomer at Γ ∗ obtained from the scale analysis in thedilute-semidilute region (see Refs. [14,18] for details). Thecalculated values for z increase monotonically from a valueclose to 2 at low temperature to reach an asymptotic limitclose to 3.5 at room temperature. In fact, the only admis-sible values for this parameter are comprised in the rangebetween z = 2, value for a fully-extended chain, and z = 4which corresponds to a random coil [33]. The variation ob-served seems to suggest an internal disorganisation of thechains in increasing temperature from a close to linear toa random-like conformation. This seems to be compatiblewith the more expanded character of the isotherms (seeRef. [14]), since a random coil excludes more volume thanthe linear chain of equal molecular weight.

3.1.2 Thermoelastic properties. Thermal expansivity αand compression ε0 moduli

It is known that depending on its microscopic configu-ration, a polymer in 3D may either crystallize or cool

748 The European Physical Journal B

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Fig. 2. Thermal expansion coefficient α as a function of T atdifferent surface pressures Π. The arrow marks the hypothet-ical fluid-solid transition.

down to its glassy amorphous state, but in both cases thethermal expansion coefficient α = (1/V )(∂V/∂T )P , variesstrongly in crossing the transition temperature [3]. The2D homologue of this quantity can be calculated from thefollowing expression:

α =1A

(∂A

∂T

)Π

· (6)

We have calculated α as a function of T from the Π−Aexperimental isotherms [14]. Figure 2 shows that in thelow Π regime (Π > 0.5 mN/m, dilute-semidilute regime)there is a change in the thermal expansivity around14−15 ◦C. It is worth noting also the thermal inversion ofα in crossing to the concentrated regime at Π ≈ 12 mN/m(and Γ ≥ 1 mg/m2). Negative values of the thermal ex-pansivity are not a surprising result, in fact there are nu-merous elastic materials that show contraction upon heat-ing [27]. Negative thermal expansion coefficients have beenfound by X-ray measurements of the lattice parametersof crystalline and semicrystalline polymers in 3D [27–29].For example, α < 0 is found for cases in which uniax-ial orientation is high enough, such as that one found inamorphous extended polymers around or below Tg [27].As it has been pointed out by several authors [27,29,30],the thermal expansivity of a fully extended chain shouldbe negative because thermal vibrations lead to a shorten-ing of the chain. However, it is important to remark thatbecause of the atacticity of the PVAc, the solid phase spec-ulated here might be more compatible with an amorphousglass than with a crystalline phase, which should be notexpected in this system. Similar liquid-solid transition be-haviour can be observed by compressing the monolayer atconstant temperature, and in fact a similar inversion in αis observed in increasing Π (or Γ ). This point has beenexperimentally corroborated by Armstrong et al. [31] incolloidal monolayers, and used by Earnshaw et al. [32] toargue the success of KTHNY theory in insoluble mono-layers made of small surfactants.

Fig. 3. Static elasticity modulus calculated from the Π−Γ ex-perimental curves. The dashed lines indicate the limits betweenthe concentration regimes.

Also, from the experimental Π−Γ curves of refer-ence [18], it is possible to calculate the more habit-ual static elasticity modulus λ0 (or ε0 in more familiarnotation), which accounts only for hydrostatic compres-sion:

ε0 = λ0 = −A(∂Π

∂A

)T

= Γ

(∂Π

∂Γ

)T

· (7)

Figure 3 shows ε0 at different temperatures. It can beobserved that most of the increase in the elasticity takesplace within the semidiluted regime, where its variationwith temperature is almost within the experimental un-certainty. However, near the maximum of elasticity, cor-responding to the crossover to a concentrated regime(Γ ∗∗ ≈ 1 mg/m2, Π ≤ 12 mN/m), the temperature coef-ficient of ε0 becomes negative and non-negligible.

Since ε0 accounts only for static compression, the datapresented in Figure 3 allow one to test for liquid-solid tran-sitional scaling-like behaviour at T ≤ Tm in a similar wayto the one exhibited by solid monolayers made of smallsurfactants [32]:

ε0 = λ0 = λTm + CλtΨ . (8)

Figure 4a shows the T -dependence of ε0 for Γ = Γ ∗∗ forthe PVAc monolayers. Below Tm ∼ 15 ◦C, a strong in-crease of ε0 is observed. The fit to the scaling law in equa-tion (8) with Ψ = 0.37 follows quite good the experimentaldata for T < Tm using Tm = 14.9± 0.4 ◦C, λ0(T = Tm) =λ0 = 23.2±0.6 mN/m and Cλ = 7.2±0.9 mN/m. It mustbe remarked that the KTHNY theory leads to equation (8)with Ψ = 0.37. However, it is worth noting that the ex-perimental data around Tm show a smoother trend thatexpected for pure hexatic-fluid transitional behaviour.

We have also tried to fit the data with Ψ variable andTm fixed, since there is a strong correlation between bothparameters. Table 1 collects these parameters for several

F. Monroy et al.: Thermoelastic behaviour of polyvinylacetate monolayers at the air-water interface 749

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Fig. 4. (a) T -dependence of ε0 at Γ = Γ ∗∗ and verification ofscaling behaviour at T ≤ Tm: (—) Ψ = 0.52 and (- - -) Ψ = 0.37(see parameters in Tab. 1). (b) Experimental verification of theStruik’s relationship between the temperature coefficient of the2D static elasticity modulus ε0 and the thermal expansivity α(see Eq. (9)) at Γ = Γ ∗∗.

Table 1. Fitted values of the parameters in equation (8) atdifferent fixed values of Tm.

Tm ( ◦C) Cλ (mN/m) Ψ λTm (mN/m)

15.0 6.1± 1.3 0.52± 0.15 24.5 ± 1.5

14.9 6.4± 1.7 0.47± 0.20 24.1 ± 2.0

14.8 6.8± 1.8 0.41± 0.18 23.6 ± 2.0

14.5 7.2± 1.2 0.37± 0.15 23.2 ± 1.4

values of Tm. Because of the experimental uncertainty,it is evident that these fits do not allow clearly eluci-date between Ψ = 0.5 and Ψ = 0.37. A tendency towardΨ = 0.37 is detected also in decreasing weakly Tm in thefits. As it can be seen in Figure 4a, the validity range forsimple scaling is remarkably large: the data follow the scal-ing predictions within the experimental error at the low-est T , (Tm−T ∼ 15 ◦C). This might be interpreted as astrong influence of the critical fluctuations onto the elasticproperties of the solid network, even at temperatures farfrom Tm.

From the point of view of the equilibrium elasticity [7],the static elasticity modulus does not include any dissipa-tion. Consequently, the decrease of ε0 of the solid as Tincreases reflects the direct influence of the thermal ex-

pansion in elastic behaviour. In fact, positive values of αcorrespond to expanded polymer chains and more as Tincreases, consequently the energy-driven contribution tothe elasticity will vanish at high enough temperature [27].Furthermore, Struik [27] has found the following empir-ical relation between the thermal expansivity and thetemperature coefficient of the elasticity modulus:

− 1E

(dEdT

)≈ kα (9)

where k ≈ 10.5 for a variety of 3D polymeric amorphoussystems. By using the calculated values of α and the tem-perature coefficient of ε0 at the Γ ∗∗ state (close to themaximum of elasticity), calculated by numerical deriva-tion of the data in Figure 6a, we have verified the valid-ity of this relationship in the quasi-2D system consideredhere. Figure 4b shows the calculated values of the propor-tional constant k in equation (9). The agreement with theexpected value is quite good at T ≥ Tm (≈ 14−15 ◦C),while at T below the melting temperature an increasingnegative difference is clearly observed. These data cor-relate well with the picture of a rubber-like elastic filmat high T , which deviates toward a different solid stateat T ≤ Tm.

3.1.3 Nature of the solid phase. Enthalpic and entropiccontent

From the experimental isotherms, entropy ∆s and en-thalpy ∆h excesses with respect to the bare surface ofthe liquid subphase can be calculated. It has been foundthat both ∆s and ∆h decrease monotonically with T inall surface states. Particularly, states with ∆s < 0 overthe whole T range are found at Γ ≥ Γ ∗∗ (≈ 1 mg/m2),which indicates the higher degree of order of the mono-layers formed in the concentrated regime compared tothe free surface of the aqueous subphase. The formationprocess of these concentrated surface states is exothermicin nature (∆h < 0), pointing out their high thermody-namic stability. The ratio between the entropic fS and en-ergetic fE contributions to the total elastic force f , underconstant area conditions is given by [3]:

fS

fE=

T

ε0

(∂ε0

∂T

)A

1− T

ε0

(∂ε0

∂T

)A

· (10)

This ratio must correlate with the ratio between the en-ergetic and entropic contributions to the total free energyof the monolayer:

fS

fE∼ ∆s

∆h+ T∆s· (11)

The quantity in equation (10) has been calculated in thereference state Γ = Γ ∗∗ and it has been compared withthe energetic-to-entropic ratio expressed in equation (11).

750 The European Physical Journal B

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Fig. 5. Relative contribution of the energetic fE and entropicfS terms to the total elasticity, (�) calculated from the exactformula in equation (10), and (◦) from the ratio between thevariation of entropy and free energy in the monolayer formationprocess (see Eq. (11)).

Figure 5 shows the correlation between the elastic andthermodynamic calculation. In fact, a minimum appearsat the same temperature, around 13 ◦C, which could be as-sociated to the effectiveness of the entropic contribution,at temperatures just below Tm, to promote the solid-fluidtransition. Since |fS/fE| < 1, the dominance of the ener-getic contributions can be inferred. Moreover, the negativevalues of this ratio is an indirect signature of the near-to-fully-extended configuration of the polymer chains as ithas been earlier emphasised. Particularly, the statisticalmicromechanical theory of rubber elasticity developed byFlory and Rehner [33], predicts a direct relationship be-tween the energetic contribution fE and the end-to-enddistance as follows:

fE ∼ Td ln〈r2〉0

dT· (12)

In fact, in discussing the α data (see Fig. 2), we havealready mentioned that only a nearly fully-extended con-figuration allows contraction in heating the system. Onlyin this particular case, negative values of fE have beendescribed. This point has been extensively corroboratedin polymers in 3D. More precisely, by using mechanicaldilatometry, Mark [3] has obtained negative values in lin-ear polyethylene. Therefore, the observed thermoelasticityis compatible with a energy-driven mechanism, as a conse-quence of the extended configuration of the PVAc chainsat the air-water interface.

3.2 Surface viscoelasticity

Mechanical step-compression experiments (MSC) onPVAc monolayers have been carried out at different tem-peratures near the Γ = Γ ∗∗ surface state. As it has beenshown in a previous study [18], this system exhibits a

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complex viscoelastic relaxation over several decades infrequency. The experimental relaxation spectra H(ln τ)are a discrete sequence of broadened Gaussians (Maxwellmodes) with a series of characteristic relaxation times ingood agreement with the normal modes structure pre-dicted by a 2D reptation model recently proposed byNoskov [34]. At higher frequencies, by combining electro-capillary excited waves (ECW) (20 Hz–3 kHz) and SQSLS(10 kHz–200 kHz) techniques, it has been shown that a se-ries of independent diffusion modes do exist (see Ref. [18]for a more detailed description). In order to explain brieflythe reasons to conjecture transitional behaviour from thelow ω data, Figure 6 shows the reptation relaxation timesas a function of T , where it is easy to observe a cleardiscontinuity in the slowest relaxation time.

As it has already been demonstrated in reference [18],the T -dependence of these low frequency data followsthe WLF equation, and the discontinuous jump in thedirector time of the reptation dynamics τ1 is predictedat T = Tm = 14 ± 1 ◦C. Keddie et al. [5] have foundthat the Tg of thin films of PMMA and PS depositedon solid substrates is smaller than for bulk samples. Thebulk Tg of our PVAc sample is 44 ◦C, thus the transitionfound at Tm = 14 ◦C might be associated to the Tg ofthe quasi-2D system considered here. Moreover, the ob-served discontinuity at low ω progressively vanishes as

F. Monroy et al.: Thermoelastic behaviour of polyvinylacetate monolayers at the air-water interface 751

Fig. 7. T -dependence of the static limit of the dilational vis-cosity κ. Above Tm Arrhenius-like behaviour (—) with activa-tion energy Ea ∼ 62 kJ/mol is found. Below Tm a divergentpower-like behaviour is found (- - -).

the motion become faster, suggesting the no existence oftransitional fluctuations at high ω. In fact, the light scat-tering data (∼ 25 kHz) at the same surface concentrationconfirm the total disappearance of the WLF T -dependenceof the relaxation time for the high ω motion, which,in contrast, becomes Arrhenius-like with high activationenergy (Ea ∼ 87 kJ/mol) [18].

3.2.1 Dilational viscosity

Here, we are only interested in asymptotic viscoelastic be-haviour, as free as possible of relaxation effects, therefore,only the zero-frequency limit of the low ω data will bediscussed in detail. Figure 7 shows the T -dependence ofthe static limit of the dilational viscosity κ(ω → 0) cal-culated from the data in reference [18]. A divergence isobserved in κ(ω → 0) around Tm ∼ 14 ◦C. Below Tm, astrong increase with T is detected in approaching the tran-sition point. The experimental points have been fitted toan empirical power law ∼ t−β with t = (Tm − T )/Tm.Fixing Tm = 14.0 ◦C, one obtains a good quality fit withβ = 0.37 ± 0.05 as in the previous section (see Fig. 4).Above the transition, a smoother decrease of κ on increas-ing T is observed. This variation follows the Arrheniuslaw, as expected for the molten state. The calculated valuefor the activation energy Ea = 62 ± 8 kJ/mol is slightlylower than the high-ω value (Ea = 87 kJ/mol) obtainedfrom the light scattering data [18]. This means that theenergetic cost of pulling apart two contiguous frictioningelements in the monolayer is lower for the slowest mo-tion, evidencing synergism in the long-range interactions,responsible for the collective character of the reptationmolecular motions. This is compatible with the spectrumshown in Figure 8. As it can be observed, the area sub-tended by the different relaxation process increases with

Fig. 8. Frequency dependence of the dilational loss modulusωκ of a PVAc monolayer of Γ = Γ ∗∗ at 25 ◦C. The results havebeen obtained by combination of mechanical step-compression(MSC) experiments, electrocapillary waves (ECW) and surfacelight scattering (QESLS).

log ω which indicates that the total viscous dissipationgrows exponentially at high ω.

3.2.2 Dilational and shear elasticity

In order to check for solid-like elastic behaviour, it wouldbe necessary to obtain simultaneously the two dilationalelastic moduli, compression λ and shear µ. The problemis that the bulk modulus ε(ω) involved in the disper-sion equation contains these two moduli coupled together.However, and since the static elasticity modulus λ0 = ε0

only contains hydrostatic compression, it would be possi-ble to discriminate the partial contribution of the shearcomponent to the total dilational modulus if the furtherincrease in λ(ω) due to internal viscoelastic relaxation wasconsidered. For insoluble monolayers, the excess in the di-lational modulus at high frequency ε(ω) with respect tothe static value ε0, is due to two simultaneous and inde-pendent factors: relaxation and shear effects. Frequently,shearing effects are not considered because of the fluidcharacter of the more usual monolayers, but for a solidmonolayer they should be taken into account. Therefore,and assuming a Maxwell-like relaxation [35], the expectedω-functionality of the dilational modulus is given by:

ε(ω) = µ+ λ0 +Σkλkω2τ2

k

1 + ω2τ2k

(13)

where λk accounts for the amplitude of the different re-laxation process that could exist, which, if the strain issmall enough and the system is in the linear viscoelasticregime, should only produce adsorption or desorption of asmall fraction β = δΓ ′/δΓeq � 1, of the total amount ofmonomers involved in a static compression, δΓeq.

752 The European Physical Journal B

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�D�ε(

ω)�P

1�P�

� � �� �� �� ��

�

�

�

�

�

�E�

Π �P1�P�

κ(ω

)���

��

V�S��

Fig. 9. High frequency total dilational viscoelastic parametersε(ω) (shear plus compression) of PVAc monolayers at differenttemperatures: (a) elastic or store modulus ε(ω) and (b) totaldilational viscosity κ(ω).

In the case of a small uniaxial dilation θλ = δΓeq/Γeq,a total stress relaxation is therefore observed, δΠ(ω) =δΠeq(δΓeq) + δΠ ′ = ε(ω)θλ. In the linear regime, whichis the case of hydrodynamic motion induced by surfacemodes or the small deformations involved in the MSC ex-periments, one has δΠ ′ ≈ (∂Π/∂Γ )δΓ ′ = λ0(δΓ ′/Γeq),and ε(ω) ≈ λ0(1 + β). Therefore, it does not seemtoo plausible that the additional relaxations increase theelastic modulus more than the static elasticity itself.Consequently, and following the above argument, therewill be a maximum limit for the relaxation componentto the compression modulus:

Σkλk = βλ0 ≤ λ0 (14)

from which the range for the shear component can bewritten as:

µ+ = ε(ω)− 2λ0 ≥ µ ≥ µ− = ε(ω)− λ0. (15)

Therefore, at least an estimation of the shear modulus canbe given if one knows the dilational modulus at frequencieshigh enough. Figure 9 shows the results for dilational vis-coelastic modulus at high frequency (∼ 25 kHz QESLS) atsome of the measured temperatures. In order to comparesurface states with equivalent energy-driven elasticity, theΠ = constant thermodynamic path has been chosen. Atlow Π(≤ 2 mN/m), ε(ω) is small and remains constant

��

��

��

��

���

� � �� �� �� ��

ε�P

1�P�

ε(ω)λ� ε

�

� � �� �� �� ��

�

��

��

��

µ�

µ�

7 ��&�µ�P

1�P�

Fig. 10. (a) T -dependence of the elasticity modulus at Π =12 mN/m ∼ Γ ∗∗, (�) total elasticity ε(ω) = λ + µ atq = 229.6 cm−1, and (◦) static compression modulus λ0. (b)Estimated upper (µ+) and lower (µ−) limits for the shear com-ponent of the elastic modulus (see Eq. (15)).

with Π over the whole temperature range. In a similarway to the static modulus, ε(ω) grows suddenly in thesemidiluted regime until it reaches a maximum value atΠ = Π∗∗ ∼ 12 mN/m [Γ (Π∗∗) ∼ Γ ∗∗ ∼ 1 mg/m2],which is always higher than the static value at the sameΠ. However, it is worth noting that while at high T thisdifference is small, at temperature low enough, the maxi-mum elasticity value increases strongly with regard to thestatic value. At T = 1.35 ◦C one has ε0 = 27.0 mN/m andε(ω) = 83.70 mN/m, i.e., between 30 and 57 mN/m of thistotal amount (36–68%) corresponds to the shear compo-nent. The experimental values of the dilational viscosityκ, shown in Figure 9b, vary with Π (or Γ ) in a similar waythan ε(ω). Its T -dependence is less clear, remaining almostconstant within the experimental error. Figure 10 showsboth, the high-ω elasticity modulus and the static valueat Π = 12 mN/m. The range of variation of µ, calculatedfrom equation (15) have been also included. Since neg-ative values for µ− are physically meaningless they havebeen made equal to zero. The kink in the slope of the ε(ω)data in Figure 10a is less clear than for the static ones λ0,(see Fig. 4a, also) on one side due to its greater absolutevariation and, on the other side because of an eventualq-induced Tm-drift to higher temperatures. However, theconcavity of ε(ω) at T ≤ Tm (static conditions ∼ 15 ◦C),does not seem compatible with a scaling behaviour with

F. Monroy et al.: Thermoelastic behaviour of polyvinylacetate monolayers at the air-water interface 753

Fig. 11. Scaling behaviour in the total elasticity coefficientε(ω) (◦) in a melting by compression process. Comparison withthe static compression modulus λ0 (�). The continuous linerepresents the fit to the scaling law in equation (16) with theexponent Ψ = 0.37.

a critical exponent Ψ lower than unity. Figure 10b shows aclear tendency to fluid-like behaviour (µ→ 0) at tempera-tures high enough. Clearly, the data at the lowest temper-atures are compatible with a certain minimum amount ofshear component, which confirms that these states behavelike a 2D solid.

A similar behaviour should be found in a solidificationprocess carried out by an isothermal compression path. Asit has been discussed in Section 3.1.3, the PVAc monolay-ers seem to behave according to an energy-driven elasticmechanism, consequently the compression path must bea more evident via for surface freezing than the thermalcooling (entropic in nature). In practice, T must be re-placed by Π, and if the film undergoes a melting processunder expansion, the elastic modulus ε(ω) should stronglydecrease at Π = Πm. The success of equation (8) leadus to consider that the elastic coefficient might follow ascaling law in the reduce pressure, ΠR = (Π −Πm)/Πm:

ε ∼ ΠΨR (16)

with Ψ = 0.37−0.5.This correspondence has been earlier recognized by

Earnshaw et al. [32] in small surfactants monolayers,which can exhibit true hexatic mesophases at monomericspecific area and temperature low enough, and where theKTHNY theory might hence apply successfully [32]. Alsowe think that since Π, instead of A, is the generalisedforce associated to the 2D mechanical work, it would bethe rigorous mechanical equivalent of T for the isothermalcompression path.

Figure 11 shows the low Π portion of the ε−Π curveat T = 1.35 ◦C. Similar results have been found at therest of temperatures studied at T < Tm. In this figure it iseasy to see the abrupt change in the elasticity coefficientat Π = Πm. At lower pressures (Π ≤ Πm ∼ 1.24 mN/m,

in the diluted regime), the total elasticity ε(ω) values areclose to the compression ones, λ0 = ε0, and close to zeroas corresponds to a fluid phase. At Π = Πm ∼ Π∗,the total bulk modulus ε(ω) = λ(ω) + µ(ω) grows sud-denly while the static compressibility λ0 remains in a lowvalue (µ � λ ∼ 0). This is compatible with the ideaof a more organized phase at lower temperature, withthe polymer chains in a near fully-extended configura-tion. Under such conditions it would be rather difficultto deform the monolayer by compression (λ ∼ 0). Thistransitional behaviour at the crossover to the semidilutedregime (Π ≥ Πm ∼ Π∗) is compatible with the scaling lawin equation (16) with critical exponent Ψ = 0.37± 0.13 inthe proposed melting by expansion process, corroboratinghence the existence of a solid shearing phase at Π ≥ Πm

and T ≥ Tm.

4 Conclusions

The thermoelastic behaviour of PVAc monolayers hasbeen reanalysed from the rheological data already pub-lished in a earlier paper, pointing out the existence of tran-sitional behaviour around Tm ∼ 14 ◦C. Above this temper-ature a viscoelastic fluid phase exists like a 2D-gel. BelowTm, a solid-like behaviour emerges, conjecturing about aglass network made of crosslinked extended chains. In par-ticular, scaling-like behaviour has been found in the vis-coelastic coefficients of the solid phase at temperaturesbelow Tm. Singular behaviour of the bulk modulus hasbeen also evidenced in a freezing by compression process; astrong increase of the elastic modulus is observed in cross-ing from the fluid state to the solid, which points out that ashear contribution µ 6= 0 emerges over the total bulk mod-ulus ε = λ+µ, as expected for a 2D-solid. Finally, from theglobal thermoelastic equilibrium behaviour of the PVAcmonolayers, the dominance of energetic effects in the elas-tic response has been inferred, in contrast with the moreusual entropic mechanism responsible for rubber elasticityin amorphous materials. An extended-like configurationalimage, compatible with the experimental Π−A isotherms,has been tentatively conjectured for the PVAc chains atthe air-water interface. This can explain the observed ther-moelastic behaviour, particularly the thermal contraction(α < 0) found in the solid state, opening the possibilityfor solid-like behaviour; a network of defects, with a fairdegree of long-range order, in a matrix made of crosslinkedfull extended chains could be formed at low temperatureand high surface density.

F. Monroy thanks Centre d’Etudes de Physique Theorique etNucleaire (CEPHYTEN France) for financial support undercontract 98/44-Rhone Poulenc. This work was supported inpart by Fundacion Ramon Areces and by DGES under grantPB96-609.

754 The European Physical Journal B

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